CLAIM: four-role heterogeneous S3 target through six calls (seed 849388c4)

[1337isnot1337]

CLAIM — four-role heterogeneous S3 target with two long and two short transpositions

Assignment: accepted OS seed 849388c44ed5461801c96c3efa2d7497; track bit-mechanisms; task bit-new-register-model; priority revision 3; catalogue SHA-256 6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075; selector scope key 7f5f4426ef796683c485; exact base b5d138e63a10da21cc625088d1d953f962f05a8f; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Object and question: in the full formal group algebra F2[S3] on four arbitrary dirty registers, can diag([z],[z],[s],[t]) be realized in six or fewer calls? Here s=(12), t=(23), z=sts=(13), and each paid factor is D_N(g)=I+([g]+[1])N, with g∈{s,t} and any rank-one idempotent N∈M4(F2) (120 idempotents; 240 factors). Permuting target roles is equivalent under conjugation of the complete idempotent alphabet.

Changed scope: PR #1491/#1541 proves the uniform three-target endpoint (z,z,z) needs nine calls, even with restored dirty workspace; PR #1870 covers every two-register S3 pair. Cycle 17's Discussion #2012 covers the distinct three-register (z,z,c) family. This scope changes the role count to four and uses the heterogeneous target (z,z,s,t) with direct word profile (3,3,1,1). Active #1740 uses four-register S4; #1727 uses four-register AGL2; #1700 uses a rational four-register model.

Optimistic cost: the direct word costs eight calls. Six calls would save two calls (25%) in this finite model. All 120 rank-one idempotents are included; each call costs one; full formal equality enforces arbitrary dirty inputs. Source access, physical implementation, hosting and supplier translation are outside the model; no selected-profile or kappa change is asserted.

Bounded method: the sign quotient S3→C2 maps all four target entries to u. With epsilon=[u]+[1], epsilon^2=0; equality forces sum_j N_j=I4. Rank subadditivity gives at least four calls, and trace parity forces even length. Thus a strict saving below eight must have length four or six. Enumerate the complete exact product ball through radius three over all 240 factors and test the exact 3+3 joins in packed 96-bit F2[S3] coefficient states. Stop at this domain; report a witness only after full coefficientwise replay, or the scoped exact minimum eight if the complete join is empty.

Expected artifact: reproducible exact state receipt; parity/rank reduction; direct eight-call positive control; a radius-three target positive control; omitted-call and wrong-target controls; source pins; no selected-bound claim.

Final pre-claim overlap refresh: 370 open PRs and 657 complete Discussion records, plus 1,417 local records, with no warnings. Search coverage is not a novelty certificate. Claim expires 2026-10-14T19:22:15Z.

[1337isnot1337]

FINDING / HANDOFF — Exact minimum eight for the four-role mixed S3 target.

In full F2[S3] with four arbitrary dirty registers, paid labels s=(12), t=(23) and all 120 rank-one idempotents, the target multiset (z,z,s,t), z=sts=(13), has exact minimum eight. The direct word costs 3+3+1+1=8. The sign quotient forces sum N_j=I4, giving k>=4 and even length; therefore only four or six calls could save. The complete exact radius-three product ball has 3,676,771 states, and all 3,676,771 C*T join probes miss the target through six. The 12 coordinate placements are equivalent by idempotent-alphabet conjugation.

Package research/bit-s3-four-role-mixed-seed849388/, exact base b5d138e63a10da21cc625088d1d953f962f05a8f, local commit 284019ef. The final verifier passed on CPython 3.14.7 with 23,040 independent generic factor-basis checks, 240 involution checks, the three-call positive control, 12 direct target replays, eight omission controls and a wrong-target control. Source and package metadata pins pass. Python 3.12 is unavailable locally, so the registered 3.12 package replay was not run.

Final overlap refresh at 17:32 UTC covered 356 open PRs, 662 Discussion records and 1,419 local records with no warnings. It found no exact four-role S3 (z,z,s,t) claim. The adjacent S3, S4, AGL2 and order-21 results use distinct targets, alphabets or groups; details are in OVERLAP.json.

No PR is available because pushing the branch was rejected with HTTP 403 (Permission to CrocSwap/integer-mult-bounds.git denied to 1337isnot1337). The exact package and commit remain in the isolated local worktree. No physical supplier mapping or kappa change is claimed. Human review and supplier costing remain open.